3.10.30 \(\int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx\) [930]

3.10.30.1 Optimal result

Integrand size = 24, antiderivative size = 309 \[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\frac {3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac {\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac {3 \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {3 \sqrt [4]{3} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}+\frac {3 \sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e}-\frac {3 \sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e} \]

output

3/2*3^(1/4)*(-e*x+2)^(1/4)*(e*x+2)^(3/4)/e-1/2*3^(1/4)*(-e*x+2)^(5/4)*(e*x 
+2)^(3/4)/e-3/2*3^(1/4)*arctan(-1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))/e* 
2^(1/2)-3/2*3^(1/4)*arctan(1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))/e*2^(1/ 
2)+3/4*3^(1/4)*ln(3^(1/2)-(-e*x+2)^(1/4)*6^(1/2)/(e*x+2)^(1/4)+3^(1/2)*(-e 
*x+2)^(1/2)/(e*x+2)^(1/2))/e*2^(1/2)-3/4*3^(1/4)*ln(3^(1/2)+(-e*x+2)^(1/4) 
*6^(1/2)/(e*x+2)^(1/4)+3^(1/2)*(-e*x+2)^(1/2)/(e*x+2)^(1/2))/e*2^(1/2)
 

3.10.30.2 Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.49 \[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\frac {\sqrt [4]{3} \left ((1+e x) \sqrt {2+e x} \sqrt [4]{4-e^2 x^2}-3 \sqrt {2} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )-3 \sqrt {2} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )\right )}{2 e} \]

input

Integrate[Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4),x]
 

output

(3^(1/4)*((1 + e*x)*Sqrt[2 + e*x]*(4 - e^2*x^2)^(1/4) - 3*Sqrt[2]*ArcTan[( 
Sqrt[4 + 2*e*x]*(4 - e^2*x^2)^(1/4))/(2 + e*x - Sqrt[4 - e^2*x^2])] - 3*Sq 
rt[2]*ArcTanh[(2 + e*x + Sqrt[4 - e^2*x^2])/(Sqrt[4 + 2*e*x]*(4 - e^2*x^2) 
^(1/4))]))/(2*e)
 

3.10.30.3 Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.80, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {456, 60, 27, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {e x+2} \sqrt [4]{12-3 e^2 x^2} \, dx\)

\(\Big \downarrow \) 456

\(\displaystyle \int \sqrt [4]{6-3 e x} (e x+2)^{3/4}dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{2} \int \frac {\sqrt [4]{3} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}dx-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \int \frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}dx-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\int \frac {1}{(2-e x)^{3/4} \sqrt [4]{e x+2}}dx+\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \int \frac {1}{\sqrt [4]{e x+2}}d\sqrt [4]{2-e x}}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \int \frac {1}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \int \frac {\sqrt {2-e x}+1}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \int \frac {1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3}{2} \sqrt [4]{3} \left (\frac {\sqrt [4]{2-e x} (e x+2)^{3/4}}{e}-\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}\right )\right )}{e}\right )-\frac {\sqrt [4]{3} (2-e x)^{5/4} (e x+2)^{3/4}}{2 e}\)

input

Int[Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4),x]
 

output

-1/2*(3^(1/4)*(2 - e*x)^(5/4)*(2 + e*x)^(3/4))/e + (3*3^(1/4)*(((2 - e*x)^ 
(1/4)*(2 + e*x)^(3/4))/e - (4*((-(ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 
+ e*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1 
/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[2 - e*x] - (Sqrt[2]*(2 - e*x)^(1/4))/ 
(2 + e*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[2 - e*x] + (Sqrt[2]*(2 - e*x)^(1/4 
))/(2 + e*x)^(1/4)]/(2*Sqrt[2]))/2))/e))/2
 

3.10.30.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 456

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && 
EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] &&  !Integ 
erQ[n]))
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

3.10.30.4 Maple [F]

\[\int \sqrt {e x +2}\, \left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}d x\]

input

int((e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/4),x)
 

output

int((e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/4),x)
 

3.10.30.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.94 \[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=-\frac {3 \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - 3 \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - 3 i \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) + 3 i \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} {\left (e x + 1\right )}}{2 \, e} \]

input

integrate((e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/4),x, algorithm="fricas")
 

output

-1/2*(3*3^(1/4)*e*(-1/e^4)^(1/4)*log((3^(1/4)*(e^2*x + 2*e)*(-1/e^4)^(1/4) 
 + (-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))/(e*x + 2)) - 3*3^(1/4)*e*(-1/e^4 
)^(1/4)*log(-(3^(1/4)*(e^2*x + 2*e)*(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(1/ 
4)*sqrt(e*x + 2))/(e*x + 2)) - 3*I*3^(1/4)*e*(-1/e^4)^(1/4)*log(-(3^(1/4)* 
(I*e^2*x + 2*I*e)*(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))/ 
(e*x + 2)) + 3*I*3^(1/4)*e*(-1/e^4)^(1/4)*log(-(3^(1/4)*(-I*e^2*x - 2*I*e) 
*(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))/(e*x + 2)) - (-3* 
e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)*(e*x + 1))/e
 

3.10.30.6 Sympy [F]

\[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\sqrt [4]{3} \int \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4}\, dx \]

input

integrate((e*x+2)**(1/2)*(-3*e**2*x**2+12)**(1/4),x)
 

output

3**(1/4)*Integral(sqrt(e*x + 2)*(-e**2*x**2 + 4)**(1/4), x)
 

3.10.30.7 Maxima [F]

\[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\int { {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} \,d x } \]

input

integrate((e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/4),x, algorithm="maxima")
 

output

integrate((-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2), x)
 

3.10.30.8 Giac [F(-2)]

Exception generated. \[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.10.30.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx=\int {\left (12-3\,e^2\,x^2\right )}^{1/4}\,\sqrt {e\,x+2} \,d x \]

input

int((12 - 3*e^2*x^2)^(1/4)*(e*x + 2)^(1/2),x)
 

output

int((12 - 3*e^2*x^2)^(1/4)*(e*x + 2)^(1/2), x)